2. The intersection of the proportional function y=2x and the hyperbola y=k/x is A(2, m), and the coordinates of a are ﹙2,4﹚.
The inverse proportional function relation is y = 8/X. The coordinates of the other intersection point are -2, -4.
It is known that y+ 1 is inversely proportional to x-3, and when x=4, y=2.
(1) Find the functional relationship between y and x.
Solution: ∫y+ 1 is inversely proportional to x-3.
∴Y+ 1=K/﹙X-3﹚
Substituting x=4 and y=2 into y+ 1 = k/(x-3) gives:
The functional relationship between 3=K ∴y and x is: y = 3/x.
(2) When x=5, y = 3/5.
(3) The relationship is an inverse proportional function.
4. (Detailed process of this problem) The image of the linear function y=ax+b intersects with the inverse proportional function A (-4,2 2) B (2 2,n), and the X axis intersects with C.
(1) Find the expression of inverse ratio and linear function.
Solution: Let the expression of inverse proportional function be y = k/x.
∵ A (-4,2) is on the inverse proportional function image.
∴ substitute a (-4,2) into y = k/x;
2=K/-4，
K=-8
The expression of inverse proportional function is y =-8/x.
And ∵B(2, n) on the inverse proportional function image,
∴ Substitute B(2, n) into y =-8/x to get:
∴n=-8/2=-4∴B(2,-4)
∵ both a and b are on the image of linear function y = ax+b.
∴ Substitute A ~-4,2 ~ B ~ 2,4 ~ into y=ax+b to get:
﹛2=-4a+b,-4=2a+b﹜
∴﹛a=- 1,b=-2﹜
∴ The expression of linear function is: Y=-X-2.
And the ∵ linear function intersects the x axis at point C.
When Y=0, -x-2 = 0 and x =-2.
∴C(-2,0)
(2)s△aob=s△aoc﹢s△cob=﹙2×2÷2﹚+﹙2×4÷2﹚=6。
(3) When the linear function is less than the inverse proportional function, the value range of X.
When -4￠X￠0 or X￠2, the linear function is smaller than the inverse proportional function.